The present invention relates to optical computers implemented as matched clutter filters, multipliers, dividers, correlators, power spectrum analyzers, conjugate transformers, convolvers and optical computers which compute the impulse response h, transfer function H, coherence function .gamma., impulse coherence .GAMMA., product S.sub.y H.sub.r, inversion 1/S.sub.x, division S.sub.y /S.sub.x, cross-correlation R.sub.yx, cross-power spectrum G.sub.yx, complex conjugate S.sub.x * and convolution y*x of two signals y and x in real time. The signals y and x may be one or two dimensional signals in a radar, sonar, communications system, mapping, surveillance, reconnaissance or pattern recognition system.
The Fourier transforms F of signals y and x are given by EQU S.sub.y =F{y} EQU S.sub.x =F{x}
(1)
from which three power spectra and corresponding time correlations may be computed. There are the cross and auto power spectra and correlations ##EQU1## where the asterisk appearing over a quantity indicates a complex conjugate and F.sup.-1 is the inverse Fourier transform of the quantity indicated. The correlations and their Fourier transforms are given by ##EQU2##
Signal x is related to the signal y by the transfer function H and impulse response h ##EQU3##
In the foregoing the impulse response h and transfer function H are equivalent statements in the time and frequency domains of the relationships between the signals y and x, for example as the received and transmitted signals of a radar or communication system or as the output and input of a system under test. In some applications, however, the measurement desired is not the relationship between signals but the causality between signals. This type measurement is obtained by computing the coherence function and impulse coherence given by ##EQU4## where .gamma. is a value lying between 0 and 1. In view of equations (4), equations (5) can also be written as follows: ##EQU5## which provides an alternative method for computing the coherence function.
It is well known in the radar and communications arts that the output of a filter S.sub.y.sbsb.o is related to its input by the filter's transfer function H.sub.r EQU S.sub.y.sbsb.o =S.sub.y H.sub.r =G.sub.yx H.sub.r ', H.sub.r =S.sub.x *H.sub.r ' (7)
where S.sub.y.sbsb.o and S.sub.y represent the frequency spectra of the output signal y.sub.o and input signal y respectively. The G.sub.yx H.sub.r ' part of equation (7) is obtained by virtue of the fact that G.sub.yx =S.sub.y S.sub.x *.
The output signal y.sub.o may be obtained using any one of the following algorithms: ##EQU6## where the integrals are over all times. Thus the output of a filter can be obtained in any one of a number of ways; by direct use of the convolution integral in the first of equations (8), by the use of equation (7) to obtain the frequency spectrum S.sub.y.sbsb.o and then using the inverse Fourier transform in the second of equations (8), or by using the difference (recursive) equations in the last of equations (8). In the present disclosure we will restrict the computations to the first and second of equations (8) and use the terms "convolution integral" and "fast convolution" to distinguish between these two equations. The latter term should not be confused with the like term of the well known Cooley-Tukey method of digital computing but simply to designate the algorithm of the second of equations (8) in the present disclosure. It will be appreciated by those skilled in the optical computer art that the terms "time" and "frequency" although clear enough in the time-frequency domains of most transformations in the communications art will be used in the sense of the spatial distributions encountered in Fourier optics as well.
A filter is said to be matched when the filter transfer function in equation (7) satisfies ##EQU7## where .vertline.N.vertline..sup.2 is the power spectrum of the noise or clutter which interferes with the signal y in the filter. The output of a matched filter is obtained by using equation (9) in equation (7) ##EQU8##
Examples of matched filters may be obtained by specifying the power spectrum .vertline.N.vertline..sup.2 of the interference in equations (9) and (10); when ##EQU9## Thus when .vertline.N.vertline..sup.2 =constant, for example thermal noise, the filter is matched for thermal noise when the transfer function H.sub.r is implemented as the complex conjugate S.sub.x * of the signal x and the filter output represents the cross correlation R.sub.yx. This is the most familiar case encountered in practice and has been discussed in a number of publications, for example in chapter 9 in the book by Skolnik "Introduction to Radar Systems" McGraw-Hill 1962. Another important case arises when the interference resembles the signal itself, when ##EQU10## Thus when .vertline.N.vertline..sup.2 =G.sub.xx, the transfer function H.sub.r can be implemented in one of a number of ways as shown in the second of equations (12) and the filter output represents the impulse response h of signals y and x. This case has been discussed in a number of publications, for example in section 12.4 of Skolnik who describes a matched filter for clutter rejection and in the article by Roth "Effective Measurements Using Digital Signal Analysis" appearing in the April 1971 issue of IEEE Spectrum. Yet another interesting case arises when the interference resembles the combination of signals; when ##EQU11## Thus when .vertline.N.vertline..sup.2 =(G.sub.yy G.sub.xx).sup.1/2, the transfer function assumes the form shown in the second of equations (13) and the filter output represents the Fourier transform of the square root of the coherence function .gamma.. This case has been described by Carter et al "The Smoothed Coherence Transform" appearing in the October 1973 issue of IEEE (Lett) Proceedings. In the present disclosure the term "matched filter" will be used to denote a matched filter for thermal noise for which .vertline.N.vertline..sup.2 =constant while the term "matched clutter filter" will denote a matched filter for clutter for which .vertline.N.vertline..sup.2 is a function of frequency.
From the foregoing it can be concluded, first, that once the nature of the interference is specified the matched filter is known, second, the filter can be implemented in any one of a number of ways using equations (8) and, third, the matched filter is a non-recursive (zeros only) type filter while the matched clutter filter is a recursive (zeros and poles) type filter. As a consequence, it is to be expected that the matched filter is a simple apparatus based on R.sub.yx and G.sub.yx while the matched clutter filter is a complex apparatus based on h and H or .GAMMA. and .gamma..
The matched filter based on R.sub.yx and G.sub.yx is useful in many practical applications especially where there exists little or no interference except thermal noise and signal y almost identically therefore resembles signal x. The matched clutter filter based on h and H is useful when the interference resembles signal x and signal y is a complex signal, for example a group or plurality of closely spaced overlapping signals each signal in the group being almost identical to signal x. The matched clutter filter based on .GAMMA. and .gamma. is useful when the interference resembles the product of signals y and x, for example when both signals y and x have been mixed.
The problem at hand is to obtain a better measurement of the time delay and frequency relationships of signals y and x in a clutter environment. Such measurements are needed in applications involving the arrival of multiple closely spaced and overlapping signals y following transmission of a signal x, for example in radar, sonar, and communications applications and in applications involving the frequency response of a system under test, for example a communication line, an amplifier and so forth. In such applications the measurement of the impulse response h and its transfer function H ##EQU12## have better time resolution and frequency response than the cross correlation R.sub.yx and its power spectrum G.sub.yx ##EQU13##
The better measurements afforded by equation (14) over equation (15) are obtained by dividing the cross power spectrum G.sub.yx by the auto power spectrum G.sub.xx or, alternatively in view of equation (4), by dividing the frequency spectrum S.sub.y by the frequency spectrum S.sub.x. This is the problem discussed both by Skolnik and Roth. It has also been suggested ad hoc by Carter et al that an even better result is obtained by dividing the cross power spectrum G.sub.yx by the square root of the product of auto correlations (G.sub.yy G.sub.xx).sup.1/2. As discussed previously, the whitening process of dividing the cross power spectrum G.sub.yx by the power spectrum .vertline.N.vertline..sup.2 of the interference results in a matched filter for the particular type of interference which is being specified in the matching.
The benefits which are to be derived from the measurement of the impulse response h, transfer function H, and coherence function .gamma. are threefold; first, it becomes possible to unambiguously determine the time delay between signals even though the signals may have complex shapes and forms, components, codings, close arrival spacings of components and overlappings, second, it becomes possible to accurately determine the performance of a system under test and, third, it becomes possible to determine the effect of noise.
In general, computations of the convolution integral of the first of equations (8) can be made using general purpose digital or analog computers or using special purpose hardware which offer significant savings in computational speeds and costs in a large number of applications. However, while the design of a matched filter involves the relatively simple problem of designing a filter having no poles and only zeros, the corresponding design of a matched clutter filter involves the increasingly difficult problem of designing a filter having both poles and zeros and this reflects directly in the weight, size, power consumption, and cost of both the hardware (analog or digital) and software which may be used. Matched clutter filters are therefore inherently more complex and costly devices when compared to simple matched filters and for this reason are not generally available for mass consumption and use. In fact the design of a matched clutter filter for real time operation becomes almost prohibitive since a large amount of paralleling of elemental hardware building blocks becomes necessary in order to achieve the desired speedup of the signal processing throughput. One feature of the optical computer is its inherent paralleling of a large number of channels. Thus, while non-optical computers increase in size, weight, power consumption, and cost quite rapidly when called upon to simultaneously process a large number of parallel channels the optical computer accomplishes this same task naturally at very high speed and thereby permits the processing of enormous amounts of information and data at the lowest possible cost.
What is important in the decision to implement a matched clutter filter is the accuracy and ambiguity which can be tolerated in the desired result. As example, many applications can be satisfied with a simple matched filter comprising a single correlator and a single Fourier analyzer to obtain the cross correlation R.sub.yx and cross power spectrum G.sub.yx from which the relationship between signals y and x may be obtained to within some low but tolerable accuracy and ambiguity. If higher accuracy and less ambiguity are desired in the application then a complex matched filter must be implemented comprising perhaps a number of correlators and Fourier analyzers to obtain the impulse response h and transfer function H. In practical terms the desire for higher accuracy and less ambiguity requires the whitening process of dividing the cross power spectrum G.sub.yx by the auto power spectrum G.sub.xx as discussed in the article by Roth or, in some applications, dividing the cross power spectrum G.sub.yx by the square root of the product of auto power spectra (G.sub.yy G.sub.xx).sup.1/2 as discussed by Carter et al. Thus the accuracy and ambiguity resolution which is required in a given application will determine the degree and type of whitening which is required in the application and consequently will determine the complexity of the apparatus which is to be used. In general, the measurement of the impulse response h based upon the whitened cross power spectra G.sub.yx /G.sub.xx or G.sub.yx /(G.sub.yy G.sub.xx).sup.1/2 is a more complex measurement than is the measurement of the cross correlation R.sub.yx based upon the unwhitened cross power spectrum G.sub.yx and consequently the apparatus of the matched clutter filter is more complex than that for the matched filter.
Once the selection of the whitening process is made in a given application the problem reduces to the implementation of apparatus having the highest possible speed and lowest possible weight, size, power consumption and cost. In general the transforms represented by equations (8) present an excessive computational load for a general purpose computer and a heavy load even for a digital computer structured for signal processing. For example, a straightforward linear transformation in a computer that takes a sequence of N data points into a sequence of N transform points may be regarded as a multiplication by a vector N.sup.2 matrix. A direct implementation therefore requires N.sup.2 words of storage and N.sup.2 multipliers (simultaneous multiplications). However it is well known that in a correlation or convolution integral one can take advantage of the fast Fourier transform algorithm (FFT) which requires only about Nlog.sub.2 N calculations instead of N.sup.2 and for N large the time and storage space saved becomes quite significant.
From the foregoing it is clear that making the needed computations using digital computers offers the potential benefit of high speed and high throughput signal processing but while this is easily said it is not easily done. For example, satellite mapping, surveillance and reconnaissance data is routinely collected over vast regions of the earth's surface providing enormous amounts of data that must be analyzed and interpreted. Both tasks have not been completely automated to provide results in real time and are accomplished primarily by skilled analysts and interpreters. The fact is that clutter filters are complex and costly devices and have not found extensive use in practice. Thus while the present art has the potential it has failed to provide a simple and economic method and apparatus for implementing clutter filters, for example for computing the impulse response h, transfer function H, coherence function .gamma. and impulse coherence .GAMMA..
It is a well known fact that the analog computer offers significant advantages in certain fields over the digital computer. For example, the analog computer offers the user low-precision but high-speed one-dimensional or two-dimensional linear discriminant analysis with a significant advantage in hardware performance (equivalent bits per second per dollar) over the digital computer in certain limited but extremely important areas. These areas include fingerprint identification, word recognition, chromosome spread detection, earth-resources and land-use analysis, and broad-band radar analysis. In these certain limited cases, defined primarily when the pattern recognition tasks require the correlation detection of features by matched filtering (linear discrimination), it may be advantageous to use the analog computer. The same is true when performing detection by means of quadratic discrimination. In such cases analog computer hardware has a significant speed advantage over most digital hardware. In some cases a considerable cost advantage may also be realized. This is particularly true in the processing of two-dimensional data where optical analog computation may be used to advantage. In addition to analog computers using optical excitation, the electronic analog computer and analog computers using acoustical excitation are well known in the prior art.
Pattern recognition by matched filtering is feasible, using optical analog computation, because of the Fourier relationship which exists between the front and backplanes of a lens. The simplest operation which can be performed by an optical analog computer is the computation of the Fourier transform S.sub.y (x.sub.x, y.sub.2) of an input pattern y(x.sub.1, y.sub.1) where y(x.sub.1, y.sub.1) is the complex signal (amplitude and phase) of the radiation in the front plane P.sub.1 of the lens and S.sub.y (x.sub.2, y.sub.2) is the complex Fourier transform of y in the backplane P.sub.2 of the lens and where x.sub.1, y.sub.1 and x.sub.2, y.sub.2 are spatial coordinates in planes P.sub.1 and P.sub.2 corresponding to the more familiar time and frequency coordinates encountered more frequently in the non-optical communication art. When the transform S.sub.y (x.sub.2, y.sub.2) is sensed by an energy detector, the result is a measure of the Wiener pattern G.sub.yy (x.sub.2, y.sub.2)=S.sub.y (x.sub.2, y.sub. 2) S.sub.y *(x.sub.2, y.sub.2) where S.sub.y * is the complex conjugate of S.sub.y. Only a single lens plus an output detector array is required to construct and record G.sub.yy. This rather elementary hardware is all that is required for implementing certain simple but very significant pattern recognition tasks including chromosome spread location and remote sensing applications. In addition to computation of the Fourier transform, the optical analog computer may be used for both frequency-domain (plane P.sub.2) and time-domain (plane P.sub.1) analysis and detection in pattern recognition systems. If, instead of an energy-detector for forming G.sub.yy, a transparency or other equivalent light modulator is placed in the frequency plane (P.sub.2) of the lens and is so structured that it is the estimate for the complex conjugate S.sub.x * of the transform S.sub.x of a pattern x related to pattern y to be identified then the product S.sub.y S.sub.x * which is formed in the frequency plane of the lens may be transformed by a second lens to produce the correlation function R.sub.yx (x.sub.3, y.sub.3) appearing in the backplane P.sub.3 of the second lens. Only two lenses plus an output detector array is required to construct and record R.sub.yx, i.e., for implementing a correlator or matched filter. And, it will be appreciated that, for implementing a convolver, the direction of inserting transparency S.sub.x * into the frequency plane must be reversed. This rather elementary hardware is all that is required in implementing certain simple but again significant pattern recognition tasks including character recognition, word recognition and broadband radar signal processing.
The main drawbacks to using optical analog computers are (1) the difficulty of input-output (I/O) conversion, (2) the inaccuracy of the computations and (3) off-line operation.
New devices for solving I/O problems include such input devices as electro-optic delay lines, membrane light modulators, and photochromic films, as well as such output devices as arrays of light detectors and television (TV) pickup tubes. These are well known in the prior art and are discussed extensively in the book by K. Preston "Coherent Optical Computers" New York, McGraw-Hill, 1972 and in the articles by B. Thompson and B. Casasent both appearing in the January 1977 Proceedings IEEE Special Issue on Optical Computing. Selection therefore of such I/O devices will be obvious to those skilled in the art; hence they will not be further discussed here.
Aberrations in the optical system limit the performance of even the most highly corrected and carefully designed optical computers. For this reason, the optical analog computer is useful where low to moderate accuracy of the computations is acceptable but extremely high-speed, high-throughput and precision are required.
The most severe limitation of the optical computer arises from the difficulty of simultaneously controlling the amplitude and phase in the frequency plane in any but a simple pattern. Interferometrically recorded frequency-plane filters while having overcome the simultaneous control of the amplitude and phase are mainly restricted as being off-line, i.e., not in real time.
In practice, the complex quantity S.sub.x *(x.sub.2, y.sub.2) may not be realized as a photographic transparency in that there is no way of producing the controllable phase modulations required or of recording of the negative values required. The matched filter must, therefore, be made by some other means. This is usually accomplished by holographic techniques where an intensity-only recording medium is placed in the frequency plane of the lens and is illuminated both by the Fourier transform S.sub.y (x.sub.2,y.sub.2) of the pattern to be recognized and by the transform S.sub.x *(x.sub.2,y.sub.2) of what is called the reference or system function. Thus while the optical computer has the potential it has the serious disadvantage of off-line or two-step holography in which the reference function S.sub.x * is mechanically recorded for placement in the Fourier or backplane of a lens.
New devices for solving the real-time operation problem include such devices as electro-acoustic, acoustic-optic devices and the electron beam-writing thermoplastic film-recording Lumatron, the von Ardenne tube, electron-beam scan laser, the Titus tube, and other devices. In some cases these devices may also be used to solve the I/O problem. These are well known in the prior art and are discussed in the article by G. Stroke "Optical Computing" appearing in the December, 1972 issue of IEEE Spectrum and in the papers by D. Casasent, H. Weiss, W. Kock, P. Greguss and W. Waidelich, and G. Winzer all appearing in the April, 1975 Special Issue on Optical Computing IEEE Transactions on Computers. Selection thereof of such real-time devices will be obvious to those skilled in the art; hence they will not be further discussed here.
The foregoing advantages and disadvantages of optical computers are well known in the prior art and can be found discussed at length in the article by K. Preston "A Comparison of Analog and Digital Techniques for Pattern Recognition" appearing in the October, 1972 issue of Proceedings of the IEEE, in the article by G. Stroke, and in papers by various authors appearing in the 1975 and 1977 IEEE Special Issues on Computers.
From the foregoing it is clear that the major impediments to the realization of many optical computing devices and systems that exhibit the full throughput and computing power possible in a (parallel) optical computer (processor) have been the realization of workable and economical real-time I/O devices and matched spatial filters capable of operating in real-time. Moreover, the real-time problem when compounded together with the inherent complexity of implementing a clutter filter, whether as an optical device or not, have prevented the optical computer from being considered for many important two-dimensional applications. Its commercial use today is out of the question and it is confined to the laboratory. Clearly, however, the clutter filter excels over the matched filter since it produces the impulse response h while the latter produces the correlation R.sub.yx of signals y and x. It will be appreciated that the significance of having an optical computer responding as the impulse response h rather than the correlation R.sub.yx is the optical computers high-speed and high-throughput and thereby providing means for processing h over the significantly less demanding computation of R.sub.yx and, as a consequence, achieving a signal pattern or picture without ambiguity or blurring. While the prior optical art suggests method and apparatus for implementing an on-line (real time) optical computer using a division filter in the Fourier plane of a lens, i.e., realizing the second of equations (8), this is done using relatively inefficient optical-to-optical (O/O) spatial light modulators (SLMs) and therefore has not succeeded in bringing forth practical and economical optical computers. On the other hand, the prior optical art nowhere suggests method and apparatus for implementing an on-line optical computer by realizing the convolution integral, first of equations (8).
From this discussion it is clear that in the past the implementation of an optical computer for the measurement of the impulse response h, transfer function H, coherence function .gamma., and impulse coherence .GAMMA. has not been attempted being restricted by the realization of even elementary on-line systems and for the inherent complexity of implementing the impulse response h over the lesser complexity of implementing the correlation R.sub.yx. As a consequence, clutter filters for many demanding and sophisticated signal processing tasks encountered in a variety of applications are only now being attempted using other than optical means and therefore not benefitting from the full potential of optical computation. In all but a few cases do such non-optical means operate on-line. For all practical purposes, although offering the highest speed, throughput, size, weight, power consumption and cost, the optical computer has not been implemented except in simple tasks inside the laboratory and in no case as an on-line clutter filter.
Therefore it is an object of the present invention to provide a method and apparatus for optically computing the impulse response h, transfer function H, coherence function .gamma., and impulse coherence .GAMMA. of a pair of signals y and x in real time.
It is also an object of the invention to provide a method and apparatus for optically computing the correlation R.sub.yx and cross power spectrum G.sub.yx of a pair of signals y and x in real time.
It is also an object of the invention to provide a method and apparatus for an optical computer based on fast convolution, using the second of equations (8).
It is also an object of the invention to provide a method and apparatus for an optical computer based on the convolution integral, using the first of equations (8).
Within the context of the foregoing objects, it is a special object of the invention to provide a method and apparatus for an efficient optical-to-optical spatial light modulator (SLM) which can be used in the invention filters and computers.
It is a further special object of the invention to provide a method and apparatus for optically computing the multiplication, inversion, complex conjugate, division and convolution of signals in real time.
It is a further special object of the invention to synthesize a number of optical elements capable of performing optical computations in an optical computer in real time.
It is another special object of the invention to provide a method and apparatus for an on-line optical computer which can be operated as a matched filter, matched clutter filter, correlator, and convolver.
It is yet another special object of the invention to illustrate a variety of configurations of an on-line optical computer implemented as a clutter filter.